# Question 13.3: Use Table 13.1 to integrate the following functions: (a) x^4......

Use Table 13.1 to integrate the following functions:

(a) $x^4$

(b) $\cos k x$,  where k is a constant

(c)  $\sin (3 x+2)$

(d) 5.9

(e) $\tan (6 t-4)$

(f)  $\mathrm{e}^{-3 z}$

(g)  $\frac{1}{x^2}$

(h) $\cos 100 n \pi t,$  where n is a constant

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.

(a) From Table 13.1, we find  $\int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c, n \neq-1 \text {. }$  To find  $\int x^4 \mathrm{~d} x \text { let } n=4$  we obtain

$\int x^4 d x=\frac{x^5}{5}+c$

(b) From Table 13.1, we find  $\int \cos (a x) \mathrm{d} x=\frac{\sin (a x)}{a}+c$.  In this case a = k and so

$\int \cos k x d x=\frac{\sin k x}{k}+c$

Note that a, b, n and c are constants. When integrating trigonometric functions, angles must be in radians.

(c) From Table 13.1, we find  $\int \sin (a x+b) \mathrm{d} x=\frac{-\cos (a x+b)}{a}+c$.  In this case a = 3 and b = 2, and so

$\int \sin (3 x+2) d x=\frac{-\cos (3 x+2)}{3}+c$

(d) From Table 13.1, we find that if k is a constant then  $\int k \mathrm{~d} x=k x+c .$  Hence,

$\int 5.9 d x=5.9 x+c$

(e) In this example, the independent variable is t but nevertheless from Table 13.1 we can deduce

$\int \tan (a t+b) \mathrm{d} t=\frac{\ln |\sec (a t+b)|}{a}+c$

Hence with a = 6 and b = −4, we obtain

$\int \tan (6 t-4) \mathrm{d} t=\frac{\ln |\sec (6 t-4)|}{6}+c$

(f) The independent variable is z but from Table 13.1 we can deduce  $\int \mathrm{e}^{a z} \mathrm{~d} z=\frac{\mathrm{e}^{a z}}{a}+c$.  Hence, taking a = −3 we obtain

$\int \mathrm{e}^{-3 z} \mathrm{~d} z=\frac{\mathrm{e}^{-3 z}}{-3}+c=-\frac{\mathrm{e}^{-3 z}}{3}+c$

(g) Since  $\frac{1}{x^2}=x^{-2}$,  we find

$\int \frac{1}{x^2} \mathrm{~d} x=\int x^{-2} \mathrm{~d} x=\frac{x^{-1}}{-1}+c=-\frac{1}{x}+c$

(h) When integrating  $\cos 100 n \pi t$  with respect to t, note that  $100 n \pi$  is a constant. Hence, using part (b) we find

$\int \cos 100 n \pi t \mathrm{~d} t=\frac{\sin 100 n \pi t}{100 n \pi}+c$

 Table 13.1 The integrals of some common functions. $f(x)$ $\int f(x) d x$ $f(x)$ $\int f(x) \mathrm{d} x$ k, constant kx + c $\cos (a x+b)$ $\frac{\sin (a x+b)}{a}+c$ $x^n$ $\frac{x^{n+1}}{n+1}+c \quad n \neq-1$ $\tan x$ $\ln |\sec x|+c$ $x^{-1}=\frac{1}{x}$ $\ln |x|+c$ $\tan a x$ $\frac{\ln |\sec a x|}{a}+c$ $\mathrm{e}^x$ $\mathrm{e}^x+c$ $\tan (a x+b)$ $\frac{\ln |\sec (a x+b)|}{a}+c$ $\mathrm{e}^{-x}$ $-\mathrm{e}^{-x}+c$ $\operatorname{cosec}(a x+b)$ $\frac{1}{a}\{\ln \mid \operatorname{cosec}(a x+b)$ $\mathrm{e}^{a x}$ $\frac{\mathrm{e}^{a x}}{a}+c$ $-\cot (a x+b) \mid\}+c$ $\sin x$ $-\cos x+c$ $\sec (a x+b)$ $\frac{1}{a}\{\ln \mid \sec (a x+b)$ $\sin a x$ $\frac{-\cos a x}{a}+c$ $+\tan (a x+b) \mid\}+c$ $\sin (a x+b)$ $\frac{-\cos (a x+b)}{a}+c$ $\cot (a x+b)$ $\frac{1}{a}\{\ln |\sin (a x+b)|\}+c$ $\cos x$ $\sin x+c$ $\frac{1}{\sqrt{a^2-x^2}}$ $\sin ^{-1} \frac{x}{a}+c$ $\cos a x$ $\frac{\sin a x}{a}+c$ $\frac{1}{a^2+x^2}$ $\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$

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